Interatomic Coulombic Decay Widths of Helium Trimer: a Diatomics-In

Interatomic Coulombic decay widths of helium trimer: A diatomics-in-molecules approach Nicolas Sisourat, Sévan Kazandjian, Aurélie Randimbiarisolo, and Přemysl Kolorenč

Citation: The Journal of Chemical Physics 144, 084111 (2016); doi: 10.1063/1.4942483 View online: http://dx.doi.org/10.1063/1.4942483 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/144/8?ver=pdfcov Published by the AIP Publishing

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Interatomic Coulombic decay widths of helium trimer: A diatomics-in-molecules approach Nicolas Sisourat,1 Sévan Kazandjian,1 Aurélie Randimbiarisolo,1,a) and Přemysl Kolorenč2 1Sorbonne Universités, UPMC University Paris 06, CNRS, Laboratoire de Chimie Physique Matière et Rayonnement, F-75005 Paris, France 2Charles University in Prague, Faculty of Mathematics and Physics, Institute of Theoretical Physics, V Holešoviˇckách 2, 180 00 Prague, Czech Republic (Received 17 December 2015; accepted 9 February 2016; published online 25 February 2016)

We report a new method to compute the Interatomic Coulombic Decay (ICD) widths for large clusters which relies on the combination of the projection-operator formalism of scattering theory and the diatomics-in-molecules approach. The total and partial ICD widths of a cluster are computed from the energies and coupling matrix elements of the atomic and diatomic fragments of the system. The method is applied to the helium trimer and the results are compared to fully ab initio widths. A good agreement between the two sets of data is shown. Limitations of the present method are also discussed. C 2016 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4942483]

I. INTRODUCTION phenomenon. Several computational approaches have been developed and used to calculate ICD widths.26–29 Among Resonance phenomena appear in many physical, chem- them, methods based on Complex Absorbing Potential (CAP) ical, and biological processes,1 for example, in atom- provide only total widths. Partial widths can be obtained molecule or electron-molecule collisions, photoionization, and using Wigner-Weisskopf method. However, this approach is autoionization events. Even though resonance states have been based on the lowest-order perturbation theory and thus gives investigated for decades,2 computing their properties is still a only estimates for the widths. To date, only the Fano- Al- challenging task. gebraic Diagrammatic Construction (Fano-ADC) method30–32 Among the resonance phenomena, the Interatomic provides reliable partial ICD widths which are necessary Coulombic Decay (ICD) effect has attracted considerable for full theoretical description of the process. This ab attention in the last decade. ICD is an ultrafast non-radiative initio method is, however, computationally expensive and electronic decay process for excited atoms or molecules not applicable to really large systems. embedded in a chemical environment, like in a cluster. Via We report here on a simpler approach based on this process, the excited species can get rid of the excess the diatomics-in-molecules (DIM) technique33 for the energy, which is transferred to one of the neighbors, and computations of the ICD widths in rare-gas clusters. In the ionizes it. The ICD process was predicted in the late 1990s by standard DIM approach, the Hamiltonian matrix elements of Cederbaum et al.3 It was experimentally demonstrated about a system are evaluated using the energies of the atoms and 10 years ago on the example of neon clusters by Marburger all pairs of atoms forming the system. The DIM Hamiltonian et al.4 and Jahnke et al.5 Since then, it was shown that ICD is matrix is generally small and scales linearly with the number a general process, taking place in a large variety of systems, of atoms within the system. Furthermore, it requires only like hydrogen bonded6,7 or van der Waals clusters.8–11 It has accurate energies of atoms and diatomic molecules which can been observed mainly after photoionization or photoexcitation nowadays be obtained with advanced and reliable quantum but has also been experimentally demonstrated after an chemistry methods. DIM methods have been successfully electron12 and ion impact.13–15 ICD was first predicted and employed for describing small molecules34,35 as well as pure observed after ionization in the inner-valence shell, but it and doped rare-gas clusters.36–39 Here, in the spirit of the was also demonstrated (i) after two-electron processes like DIM, we propose to compute the ICD widths of rare-gas simultaneous ionization-excitation16,17 and double ionization clusters from the widths of each pair of atoms forming the or excitation18,19 and (ii) in cascades after Auger20–23 and cluster. resonant Auger.24,25 In SectionII, we give the general formulas for the The energy width of a resonant state is an important computation of resonance widths from the projection-operator property to characterize the resonance. General quantum formalism. The standard DIM approach for the calculation of mechanical equations for computing the decay widths are potential energy surfaces is then presented. The DIM-based known but are only applicable to small systems. The main method proposed here for the computation of the ICD widths difficulty stems from the fact that one has to account both is finally reported. In Section III, we apply this method to for the many-body and for the scattering aspects of the decay the helium trimer and derive explicit DIM formulas for this system. ICD widths from the DIM approach are compared to a)[email protected] fully ab initio (Fano-ADC) results.

II. ICD WIDTHS IN THE PROJECTION-OPERATOR-DIM Each fragment part Hˆ α contains all kinetic energy operators FRAMEWORK and intra-atomic potential energy terms in Hˆ e which depend only on the coordinates of the electrons assigned originally to The general formulas for the calculation of the ICD α Hˆ αβ widths are derived from the theory of resonances of Fano atom . The operator contains all kinetic energy terms as and Feshbach.40,41 The wavefunctions and coupling matrix well as intra-atomic and inter-atomic potential energy terms α β elements involved in such calculations are then computed associated to the diatomic molecule . within the DIM approach. The main advantage of the latter The DIM basis set is then constructed as antisymmetrized and spin-adapted products of the atomic eigenfunctions χα is that only energies and coupling matrix elements of the m with energy eα of Hˆ α, | ⟩ atomic and diatomic fragments are needed. Computations of m resonance widths in the case of electron-molecule collisions N anti ˆ ˆ n have been performed with such combined approach (so- Ψm = A Ψm = Asˆ χm . (6) called generalized diatomic-in-molecules approach in this | ⟩ | ⟩ n=1 | ⟩ context).43,44 In the present study, we report a similar approach The operator Aˆ ensures the antisymmetrization ands ˆ creates applied to the calculations of ICD widths. spin-adapted linear combination of the atomic eigenfunction products. In order to evaluate the Hamiltonian matrix elements in the DIM basis set, the diatomic Hamiltonians Hˆ αβ are A. The projection-operator approach αβ expressed via their eigenvalues ϵ i and eigenfunctions A general description of the Fano-Feshbach theory of αβ { } ψi , resonances within the projection-operator approach is given in {| ⟩}  ˆ αβ αβ αβ αβ Ref. 42. We summarize here the results for a single resonance H = ψi ϵ i ψi . (7) state. The generalization for a non-interacting manifold of i | ⟩ ⟨ | resonance states is straightforward. The resonance is described Noting that Aˆ and Hˆ αβ commute, the operation of Hˆ αβ on by a discrete state Ψd , represented by a square integrable Ψanti is simply | ⟩ m wave function, with the energy expectation value given | ⟩  ˆ αβ anti ˆ ˆ αβ ˆ αβ αβ αβ by H Ψm = AH Ψm = A Ψi ϵ i Ψi Ψm , (8) | ⟩ | ⟩ i | ⟩ ⟨ | ⟩ ˆ Ed = Ψd He Ψd , (1) Ψαβ ⟨ | | ⟩ where i is the spin-adapted product of the diatomic | ⟩ αβ where Hˆ e is the full electronic Hamiltonian. This discrete state wavefunction ψi and the wavefunctions of electrons that | ⟩ αβ is embedded into and coupled to a continuum of final states, are not included in Hˆ , Φ Ψ  which can be written as f k = f k . Here, the decay Ψαβ αβ n | ⟩ | ⟩| ⟩ i = sˆ ψi χi . (9) channel Ψf is the eigenstate of the system after the decay | ⟩ | ⟩ | ⟩ | ⟩ n,α, β with energy Ef and k is a single-electron continuum state. | ⟩ αβ αβ Neglecting the inter-channel coupling, the coupling between The overlap element Ψi Ψm = Bim can generally be ⟨ | ⟩ 34,36 the decaying state Ψd and the final state Φf k is determined by symmetry consideration. Using Eqs. (5), | ⟩ | ⟩ (6), and (8), the DIM Hamiltonian in the matrix form is Vd, f k = Ψd Hˆ e Φf k . (2)    ⟨ | | ⟩ HDIM = Bαβ†ϵ αβ Bαβ − N − 2 eα, (10) The partial ICD width for a given channel f is then given by α β ( ) α αβ α 2 2 where ϵ and e are diagonal matrices built with the diatomic Γf = 2π Vd, f k δ Eres − k /2 − Ef , (3) | | αβ and atomic fragment energies, respectively, and B is the where Eres = Ed + ∆ Eres stands for the real energy of the overlap matrix. The electronic states of the system are then resonance; ∆ E is the( so-called) level shift. The value of k is obtained by diagonalization of the DIM Hamiltonian matrix. ( ) fully determined by the knowledge of Ed and Ef . The k index is thus dropped in the following. The total width is the sum of the partial ones over all decay channels, C. ICD widths in the DIM approach  We consider here a cluster constituted of N rare-gas Γtot = Γf . (4) (Rg) atoms in which ICD is triggered by ionization. The f system is left in one or a manifold of excited states which lie above the double ionization threshold. This is B. The diatomics-in-molecules approach typically achieved by ionization in the inner-valence shell or simultaneous ionization-excitation of outer-valence electrons. Ψ Ψ The states d , f , and thus Vd, f are computed within This is summarized as the DIM framework.| ⟩ | In⟩ this approach, the Hamiltonian of Ionization +∗ ICD + + a N-atomic system is subdivided into atomic and diatomic RgN −→ Rg −RgN−1 −→ Rg −Rg −RgN−2 , (11) parts, ( ) ( ) where Rg+∗ illustrates that the ion is in an electronic excited N−1 N N state. The last term indicates that the cluster ends with two   αβ  α Hˆ e = Hˆ − N − 2 Hˆ . (5) charges which may lead to two atomic ions or to larger ionic α=1 β>α ( ) α=1 fragments.

The manifold of non-interacting resonant states Ψd and are given in Ref. 36. In the following, we report first the the energies are obtained in this approach by diagonalizing| ⟩ the DIM matrices adapted from these formulas for the states of +∗ DIM Hamiltonian (Eq. (10)) which is built for Rg −RgN−1 helium trimers relevant for the ICD process, namely, ionized- +∗ system. We thus have excited into the 2p shell (He 2p −He2) and doubly ionized + + ( ) d DIM † (He 1s −He 1s −He). E = UHd U , (12) ( ) ( ) 2 d The DIM basis set for the ionized-excited states ( Ψm ) used here are defined as follows: {| ⟩} where U and Ed contain the eigenvectors and eigenvalues, DIM 2 d 1 2 3 respectively. The matrix Hd is constructed as in Eq. (10) Ψ = Aˆ χ + χ χ , 1x He 2px He 1s2 He 1s2 +∗ | ⟩ | ( )⟩| ( )⟩| ( )⟩ with the energies of the (Rg−Rg) and (Rg −Rg) diatomic 2 d 1 2 3 Ψ = Aˆ χ χ + χ , and atomic fragments. 2x He 1s2 He 2px He 1s2 | ⟩ | ( )⟩| ( )⟩| ( )⟩ 2 d 1 2 3 After the decay, the system is doubly ionized. The Ψ = Aˆ χ χ χ + , 3x He 1s2 He 1s2 He 2px diagonalization of the corresponding matrix provides the states | ⟩ | ( )⟩| ( )⟩| ( )⟩ Ψ ≡ ≡ f i i f ( W) and the associated energies ( E ), where χ + and χ denote the atomic state | ⟩ | He 2px ⟩ | He 1s2 ⟩ He+ 2p /2P ( and) He 1s2/1(S ) of the atom i (Fig.1), E f = W H DIMW †. (13) x x f respectively.( ) There are( 6 similar) DIM functions (3 for the DIM y and z directions) included in the calculations. The DIM The matrix Hf is built using the energies of the neutral 1 f basis set for the singlet final states ( Ψm ) is given + + + (Rg−Rg), singly (Rg −Rg), and doubly (Rg −Rg ) ionized as {| ⟩} states of the diatomic and atomic fragments. The matrices 1Ψf = Aˆ χ1 χ2 χ3 U and W provide the Ψd and Ψf states within the 1 He+ 1s He+ 1s He 1s2 | ⟩ | ⟩ | ⟩ (| ( )⟩| ( )⟩| ( )⟩ DIM basis set, respectively. The coupling matrix between √ these states in the DIM approach has a similar form as − χ1 χ2 χ3 / 2, He+ 1s He+ 1s He 1s2 Eq. (10),43 | ( )⟩| ( )⟩| ( )⟩) 1Ψf = Aˆ χ1 χ2 χ3 2 He+ 1s He 1s2 He+ 1s    | ⟩ (| ( )⟩| ( )⟩| ( )⟩ V DIM = U Bαβ†V αβ Cαβ − N − 2 V α W †, √ d, f  d, f d, f  1 2 3  α β ( ) α  − χ + χ 2 χ + / 2,   | He 1s ⟩| He 1s ⟩| He 1s ⟩)   (14) ( ) ( ) ( )   1 f ˆ 1 2 3   Ψ = A χ 2 χ + χ + αβ  α  3 He 1s He 1s He 1s where V and V are constructed with the partial widths | ⟩ (| ( )⟩| ( )⟩| ( )⟩ d, f d, f √ of the α β diatomic and α atomic fragments (Γ , see 1 2 3 f − χ 2 χ + χ + / 2, αβ αβ He 1s He 1s He 1s Eqs. (2) and (3)). The matrices B and C are the | ( )⟩| ( )⟩| ( )⟩) overlap matrices (as in Eq. (10)) between the DIM basis where the overline indicates a spin down (β). The expression functions (Eq. (6)) and the diatomic fragment wavefunc- for the triplet final states is given by the corresponding spin tions (Eq. (9)) for the decaying and final states, respec- function. tively. For convenience, we introduce the following notation Within the DIM approach, it is thus only necessary for the helium diatomic fragment energies used in this to compute the couplings between the decaying and the work: continuum final states (Eq. (2)) of all atoms and all pairs of atoms forming the cluster. It reduces to the calculations of widths of atomic and diatomic systems which can be handled by ab initio methods already available as for instance the Fano-ADC approach. It should be mentioned that decay widths of large heteroatomic clusters have been estimated in a pairwise additivity approximation:45,46 the total width is given by the sum of the widths of all individual pairs forming the cluster. The DIM approach goes beyond this approximation since delocalization of the charges in the decaying and final states is taken into account. Furthermore, partial widths can also be computed within the DIM method.

III. APPLICATION OF THE DIM APPROACH TO ICD IN He3 A. General formulas

General formulas for the application of the DIM to FIG. 1. Geometry of the trimer: the atoms lie on the xz plane. The distance compute the PES of homogeneous noble gas ionic clusters between the atoms A and B is kept fixed at R12 = 4 Å.

S = E 1 + R , Several analytical expressions for the ground electronic i j He2, Σg i j ( )( ) S 47–49 C = E ++ 1 + R , state of neutral He2 ( ) have been proposed. However, i j He , Σg i j 2 ( )( ) the ICD widths obtained within the DIM approach are C¯i j = E ++ 3 + Ri j , He , Σu nearly independent of the neutral potential energy curve. 2 ( )( ) 1 The diatomic fragment energies used for the lowest ionized Q˜ i j = E + 2 + Ri j + E + 2 + Ri j , He , Σu He , Σg ˜ ˜ 2( 2 ( )( ) 2 ( )( )) states (G and U) are obtained from Ref. 50 and those of 1 the ionized-excited states (G, U, G¯, and U¯ ) are taken from J˜ = E + 2 + R − E + 2 + R , i j He , Σu i j He , Σg i j 2( 2 ( )( ) 2 ( )( )) Ref. 51. For the potential of the doubly ionized dimer (C and 1 C¯ Q = E +∗ 2 + R + E +∗ 2 + R , ), we used a purely Coulombic potential, which is correct in i j He , Σu i j He , Σg i j 2( 2 ( )( ) 2 ( )( )) the interatomic distance range considered here.51 1 J = E +∗ 2 + R − E +∗ 2 + R , The trimers are described in the Cs point group and the i j He , Σu i j He , Σg i j 2( 2 ( )( ) 2 ( )( )) atoms are chosen to lie in the xz-plane as shown in Fig.1. 1 ¯ +∗ 2 +∗ 2 The vector going from atom i to atom j is denoted R⃗ . For Qi j = EHe , Πg Ri j + EHe , Πu Ri j , i j 2( 2 ( )( ) 2 ( )( )) the decaying states, the DIM matrices have block structure of 1 ′ ′′ ¯ +∗ 2 − +∗ 2 × × Ji j = EHe , Πg Ri j EHe , Πu Ri j . 6 6 A and 3 3 A matrices, 2( 2 ( )( ) 2 ( )( ))

q1 + q2 + S12 p1 + p2 j2 k2 j1 k1

p1 + p2 q¯1 + q¯2 + S12 k2 j¯2 k1 j¯1

* j2 k2 Q12 + q2 + S13 p2 J12 0 + HDIM = . / (15) d 2A′ . ¯ ¯ ¯ / ( ) . k2 j2 p2 Q12 + q¯2 + S13 0 J12 / . / . j1 k1 J12 0 Q12 + q1 + S23 p1 / . / . k1 j¯1 0 J¯12 p1 Q¯ 12 + q¯1 + S23/ . / and , -

Q¯ 13 + Q¯ 23 + S12 J¯23 J¯13 HDIM J¯ Q¯ + Q¯ + S J¯ . d 2A′′ = 23 12 23 13 12 (16) ( ) * ¯ ¯ ¯ ¯ + . J13 J12 Q12 + Q13 + S23/ . / The general form of the matrices presented above is the same as in Ref. 36. However, the diatomic fragments energies , - that enter the matrix elements are that of the ionized-excited dimer states. The singlet and triplet final states can be considered separately. The corresponding DIM matrices are 3 × 3 A′ matrices. For the singlet states, the matrix reads

Q˜ 13 + Q˜ 23 + C12 J˜23 J˜13 HDIM J˜ Q˜ + Q˜ + C J˜ . f 1A′ = 23 12 23 13 12 (17) ( ) * ˜ ˜ ˜ ˜ + . J13 J12 Q12 + Q13 + C23/ . / , -

The matrix corresponding to the triplet states is obtained by decaying states is E He+ n = 2 = 65.4 eV and that of the +( ( )) replacing Ci j by C¯i j in the above matrix. The symbols used in final states is 2E He 1s = 49.1 eV. the matrices are defined as follows: The diatomic( coupling( )) matrix elements are taken from Ref. 51 and are used as follows: 2 2 qi = Q¯ i3 sin βi + Qi3 cos βi,   2 ¯ 2 1 2 + 1 + 2 + 1 + q¯i = Qi3 sin βi + Qi3 cos βi, S+ Σg → Σg Σu→ Σg Λi j = √ Γ Ri j + Γ Ri j , p = Q − Q¯ sin β cos β , 2 π ( ( ) ( )) i i3 i3 i i   ( ) 1 2 1 + 2 1 + k = J − J¯ sin β cos β , ¯ S+ Πg → Σg Πu→ Σg i i3 i3 i i Λi j = √ Γ Ri j + Γ Ri j , (¯ 2 ) 2 2 π ( ( ) ( )) ji = Ji3 sin βi + Ji3 cos βi,   1 2 + 1 + 2 + 1 + 2 2 S− Σg → Σg Σu→ Σg j¯i = Ji3 sin βi + J¯i3 cos βi, Λ = √ − Γ Ri j + Γ Ri j , i j 2 π ( ( ) ( ))   ⃗ ⃗ 1 2 1 + 2 1 + where βi defines the angle between Ri3 and R12 (see Fig.1). S− Πg → Σg Πu→ Σg Λ¯ = √ Γ Ri j − Γ Ri j , The atomic energies are not included in the above i j 2 π ( ( ) ( )) matrices. They lead to a global energy shift and do not play a role in the calculations of the widths: if the energy where the superscript S denotes singlet final states. Similar of the neutral cluster is set to zero, the energy shift of the matrix elements are defined for the triplet final states.

It should be noted that intra-atomic decay channels are matrix elements are thus set to zero. The DIM coupling energetically closed in helium clusters. The atomic coupling matrices for the singlet final states are

S+ ¯ S+ S− ¯ S− 0 Λ13 cos β1 − Λ13 sin β1 Λ23 cos β2 − Λ23 sin β2 S+ ¯ S+ S− ¯ S− 0 Λ13 sin β1 + Λ13 cos β1 Λ23 sin β2 + Λ23 cos β2 *ΛS− 0 ΛS+ cos β − Λ¯ S+ sin β + DIM . 12 23 2 23 2/ † V = U2A′ . − / W1A′ (18) 2A′,1A′ .Λ¯ S ΛS+ β Λ¯ S+ β / . 12 0 23 sin 2 + 23 cos 2/ .ΛS+ ΛS− − Λ¯ S− / . 12 13 cos β1 13 sin β1 0 / . ¯ S+ S− ¯ S− / .Λ12 Λ13 sin β1 + Λ13 cos β1 0 / . / , -

and to the complexity to evaluate accurate resonance widths, such agreement is satisfactory. ¯ S− ¯ S+ The dependence of the total decay widths with respect to 0 Λ13 Λ23 DIM ¯ S+ ¯ S− † Q (see Fig.1) is shown and compared to the Fano-ADC results V = U2A′′ Λ 0 Λ W1A′ . (19) ′ 2A′′,1A′ 12 23 in Figs.2 and3. Results for the A states (Fig.2) of the DIM *Λ¯ S− Λ¯ S+ 0 + . 12 13 / approach compare quantitatively with that of the Fano-ADC . / calculations for Q above 2-3 Å (i.e., R = 2.8-3.6 Å). For The DIM coupling matrices, for the triplet- final states smaller distances, the agreement is less satisfactory. This is are obtained by replacing in the above matrices the diatomic due to the limited DIM basis set used and the stronger 3-body fragment coupling matrix elements by that corresponding to the triplet.

TABLE I. Total decay widths at equilateral geometry (R12 = R13 = R23 B. Total decay widths = 4 Å) obtained with the DIM and Fano-ADC approaches.

−5 −5 The three helium atoms in the trimer are weakly bound ΓDIM (10 a.u.) ΓFano−ADC (10 a.u.) 52,53 such that the average interatomic distance is about 10.4 Å 2A′ 1Σ+/2P 1.21 1.46 53,54 g x and there is no preferential equilibrium geometry. In this 2 ′(1 + 2 ) A Σg / Pz 3.09 2.70 2 ′(2 + 1 ) study, we have investigated the trimer in equilateral and A Σg / S 2.89 2.67 2 ′(2 + 1 ) isosceles geometries. As shown in Fig.1, the three atoms lie A Σu/ S 2.89 2.74 2 ′(2 1 ) on the xz plane and the distance between the two atoms 1 and A Πg / S 1.29 1.49 2 ′(2 1 ) 2 is kept fixed at R = 4 Å. The distance between the center A Πu/ S 1.29 1.46 12 ( ) of mass of the two fixed atoms and the third atom is denoted 2 ′′ 1 + 2 A ( Σg / Py) 0.83 0.85 Q. 2 ′′ 2 1 A ( Πg / S) 0.80 0.83 2 ′′ 2 1 The electronic states of the trimer are labeled according A ( Πu/ S) 0.90 0.89 to their spatial symmetry classification in the Cs point group (A′ or A′′) and their spin multiplicity (doublet for the decaying states and singlet/triplet for the final states) and according also according to the symmetry of the asymptotic atomic and diatomic fragments. For 2 ′ 1 + 2 example, A Σg / Px is used for the decaying state that ( ) 1 + +∗ 2 converges at large Q to He2 Σg and He Px . The label 2 ′ 2 1 ( ) ( ) A Πu/ S is employed for the decaying state corresponding ( +∗ 2 ) 2 1 to He2 Πu –He 1s / S . The final state converging to singlet + 2 +( ) +( 2 ) 1 ′ 2 + 2 He2 Σg and He S is labeled A Σg / S and that converg- ( ) 2+ 1 +( ) 2 1 ( ) 1 ′ 1 + 1 ing to He2 Σg –He 1s / S is named A Σg / S , for instance. ( ) ( ) ( ) The total decay widths for each decaying electronic state at an equilateral geometry (R12 = R13 = R23 = 4 Å) are shown in TableI. Details on the Fano-ADC calculations and a thorough discussion of decay widths in helium trimer as well as a comparison with helium dimer data can be found in Ref. 55. The widths obtained from the DIM calculations differ FIG. 2. Total widths for the A′ decaying states. DIM and Fano-ADC results from the ab initio Fano-ADC results by at most 20%. Owing are shown in dots and in lines, respectively.

′′ 2 1 ′′ FIG. 5. Partial widths to the singlet final states for the A Πg / S decaying FIG. 3. Total widths for the A decaying states. DIM and Fano-ADC results ( ) are shown in dots and in lines, respectively. state. DIM and Fano-ADC results are shown in dots and in lines, respectively. The sharp peaks around Q = 3.5 Å are attributed to strong mixing between the decay channels near equilateral geometry.55

effects, expected at short interatomic distances, not accounted The partial widths for the A′′ (1Σ+/2P ) decaying state for by the DIM approach. However, the DIM results agree g y are well reproduced by the DIM approach. The partial well in the relevant interatomic distance range since the mean widths corresponding to the deexcitation of the He+ 2P He–He distance in clusters goes from 10.4 Å (trimer) to y and ionization of He show a 1/R6 behavior as expected( ) 3.6 Å (superfluid liquid helium).39 The DIM calculations 2 by the virtual photon exchange mechanism.16,56 The width reproduces well the widths for the A′′ states (Fig.3), even corresponding to the double ionization of the He is at at short interatomic distances. The corresponding orbitals are 2 least 2 order of magnitude smaller. This channel corresponds perpendicular to the trimer plane and therefore overlap less 57,58 ′ to Electron-Transfer-Mediated-Decay (ETMD) in which than that for the A states, which explains the better agreement. + one electron from He2 is transferred to He and a second electron from He2 is ionized. This process is efficient only at C. Partial decay widths and limit of the DIM approach short interatomic distances for which significant spatial orbital overlap is possible. We have shown that the DIM approach can be successfully ′′ 2 1 For the A Πg/ S decaying state, the DIM approach employed to compute accurate total decay widths. In order to reproduces well( the asymptotic) widths for the dominant give a full description of ICD, the partial widths to each of the channel (i.e., two-site double ionization of the He2) but fails for singlet and triplet final states must be well reproduced within the weakest ones (i.e., ionization of the third helium atom). For the DIM approach. The DIM partial widths to these states 2 ′′ 2 1 1 ′ 2 + 2 ′′ the A Πg/ S → A Σu/ S transition, the DIM underesti- for two A decaying states are compared to the Fano-ADC mates the( asymptotic) widths( by) a factor of 2. Furthermore, results in Figs.4 and5. 2 ′′ 2 1 1 ′ 2 + 2 the ab initio widths for the A Πg/ S → A Σg / S transition exhibit a 1/R10 behavior demonstrating( ) a dipole-forbidden( ) transition. Indeed, as shown in Ref. 55, the decay to these channels is asymptotically mediated by a quadrupole- quadrupole interaction. In this limit, the deexcitation of the 2 2 + dimer from Πg to Σg is dipole-forbidden. This is a 3-body effect that is not accounted for in the DIM approach and represents a limit to the calculations of partial ICD widths with the DIM method. It should, however, be noted that these channels are always weak and do not significantly contribute to the decay. Same conclusions are drawn for the partial widths of the last A′′ state and that of the A′ states (not shown). The DIM approach can therefore be used to describe ICD in rare-gas clusters.

IV. CONCLUSION A method combining the projection-operator approach ′′ 1 + 2 FIG. 4. Partial widths to the singlet final states for the A Σg / Py decaying of resonant scattering theory and the diatomics-in-molecules state. DIM and Fano-ADC results are shown in dots and in( lines, respectively.) The sharp peaks around Q = 3.5 Å are attributed to strong mixing between technique is reported. This combined approach is applied to the the decay channels near equilateral geometry.55 helium trimer for which working formulas are given. The total

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